American Institute of Mathematical Sciences

Advanced
January  2015, 11(1): 199-216. doi: 10.3934/jimo.2015.11.199

Sensor deployment for pipeline leakage detection via optimal boundary control strategies

Chao Xu 1, , Yimeng Dong 1, , Zhigang Ren 1, , Huachen Jiang 2, and  Xin Yu 3,

1. 

State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems & Control, Zhejiang University, Hangzhou, Zhejiang 310027, China, China, China
2. 

Institute of Operations Research & Cybernetics, Zhejiang University, Hangzhou, Zhejiang 310027, China
3. 

Ningbo Institute of Technology, Zhejiang University, Hangzhou, Zhejiang 310027, China

Received  December 2012 Revised  January 2014 Published  May 2014

We consider a multi-agent control problem using PDE techniques for a novel sensing problem arising in the leakage detection and localization of offshore pipelines. A continuous protocol is proposed using parabolic PDEs and then a boundary control law is designed using the maximum principle. Both analytical and numerical solutions of the optimality conditions are studied.
Keywords: Offshore pipeline network, leakage detection, optimal control., continuation approach, multi-agent system, partial differential equations, Lagrangian sensor.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Chao Xu, Yimeng Dong, Zhigang Ren, Huachen Jiang, Xin Yu. Sensor deployment for pipeline leakage detection via optimal boundary control strategies. Journal of Industrial and Management Optimization, 2015, 11 (1) : 199-216. doi: 10.3934/jimo.2015.11.199
References:
[1]

N. Ahmed and K. Teo, Optimal Control of Distributed Parameter Systems, North Holland, 1981.

[2]

S. Anita, V. Arnautu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8098-5.

[3]

V. Arnautu and P. Neittaanmaki, Optimal Control from Theory to Computer Programs, Kluwer Academic, Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3.

[4]

P. Barooah, P. Mehta and J. Hespanha, Mistuning-based control design to improve closed-loop stability margin of vehicular platoons, IEEE Transactions on Automatic Control, 54 (2009), 2100-2113. doi: 10.1109/TAC.2009.2026934.

[5]

S. Blazic, D. Matko and G. Geiger, Simple model of a multi-batch driven pipeline, Mathematics and Computers in Simulation, 64 (2004), 617-630. doi: 10.1016/j.matcom.2003.11.013.

[6]

F. Bullo, J. Cortes and S. Martinez, Distributed Control of Robotic Networks (In Applied Mathematics Series), Princeton University Press, New York, 2009.

[7]

M. Chen and D. Georges, Nonlinear optimal control of an open-channel hydraulic system based on an infinite-dimensional model, in Proceeding of the Conference on Decision and Control, vol. 5, 1999.

[8]

H. Cho and G. Hwang, Optimal design for dynamic spectrum access in cognitive radio networks under rayleigh fading, Journal of Industrial and Management Optimization, 8 (2012), 821-840. doi: 10.3934/jimo.2012.8.821.

[9]

E. Chow, L. Hendrix, M. Herberg, S. Itoh, B. Kong, M. Lall and P. Srevens, Pipeline Politics in Asia: The Intersection of Demand, Energy Markets, and Supply Routes, National Bureau of Asian Research, 2010.

[10]

Y. Ding and S. Wang, Optimal control of open-channel flow using adjoint sensitivity analysis, Journal of Hydraulic Engineering-ASCE, 132 (2006), 1215-1228. doi: 10.1061/(ASCE)0733-9429(2006)132:11(1215).

[11]

Z. Feng, K. Teo and V. Rehbock, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. doi: 10.3934/jimo.2005.1.499.

[12]

Z. Feng, K. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024.

[13]

G. Ferrari-Trecate, A. Buffa and M. Gati, Analysis of coordination in multi-agent systems through partial difference equations, IEEE Transactions on Automatic Control, 51 (2006), 1058-1063. doi: 10.1109/TAC.2006.876805.

[14]

P. Frihauf and M. Krstic, Leader-enabled deployment onto planar curves: A pde-based approach, IEEE Transactions on Automatic Control, 56 (2011), 1791-1806. doi: 10.1109/TAC.2010.2092210.

[15]

R. Glowinski, J. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, (Encyclopedia of Mathematics and its Applications) Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.

[16]

H. Hao and P. Barooah, On achieving size-independent stability margin of vehicular lattice formations with distributed control, IEEE Transactions on Automatic Control, 57 (2012), 2688-2694. doi: 10.1109/TAC.2012.2191179.

[17]

H. Hao, P. Barooah and P. Mehta, Stability margin scaling laws for distributed formation control as a function of network structure, IEEE Transactions on Automatic Control, 56 (2011), 923-929. doi: 10.1109/TAC.2010.2103416.

[18]

J. Kim, K. Kim, V. Natarajan, S. Kelly and J. Bentsman, PdE-based model reference adaptive control of uncertain heterogeneous multiagent networks, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1152-1167. doi: 10.1016/j.nahs.2008.09.008.

[19]

J. Kim, V. Natarajan, S. Kelly and J. Bentsman, Disturbance rejection in robust PdE-based MRAC laws for uncertain heterogeneous multiagent networks under boundary reference, Nonlinear Analysis: Hybrid Systems, 4 (2010), 484-495. doi: 10.1016/j.nahs.2009.11.005.

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Phaladelphia, 2008. doi: 10.1137/1.9780898718607.

[21]

Z. Lin, Distributed Control and Analysis of Coupled Cell Systems, VDM Verlag, Germany, 2008.

[22]

W. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions, Journal of Industrial and Management Optimization, 7 (2011), 291-315. doi: 10.3934/jimo.2011.7.291.

[23]

M. Liu, S. Zang and D. Zhou, Fast leak detection and location of gas pipelines based on an adaptive particle filter,, International Journal of Applied Mathematics and Computer Science, 15 (). 

[24]

M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks (In Applied Mathematics Series), Princeton University Press, New York, 2010.

[25]

T. Meurer and M. Krstic, Finite-time multi-agent deployment: A nonlinear pde motion planning approach, Automatica, 47 (2011), 2534-2542. doi: 10.1016/j.automatica.2011.08.045.

[26]

S. Moura and H. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs, in Proceeding of the Conference on Decision and Control, 2011, 921-928.

[27]

R. Murray, Recent research in cooperative control of multi-vehicle systems,, Journal of Dynamical Systems, (): 571. 

[28]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[29]

P. Parfomak, Pipeline Safety and Security: Federal Programs, Congress Research Services (CRS) Report for Congress, Washington, DC, 2008.

[30]

M. Rafiee, Q. Wu and A. Bayen, Kalman filter based estimation of flow states in open channels using Lagrangian sensing, Proceedings of the Conference on Decision and Control, (2009), 8266-8271. doi: 10.1109/CDC.2009.5400661.

[31]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks, (Communications and Control Engineering Series) Springer-Verlag, London, 2011.

[32]

A. Sarlette and R. Sepulchre, A PDE viewpoint on basic properties of coordination algorithms with symmetries, in Proceedings of the Conference on Decision and Control, 2009, 5139-5144. doi: 10.1109/CDC.2009.5400570.

[33]

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Edition, SIAM, Philadephia, 2004. doi: 10.1137/1.9780898717938.

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (Graduate Studies in Mathematics), American Mathematical Society, New York, 2010.

[35]

G. Wang and H. Ye, Leakage Detection and Localization of Long Distance Fluid Pipelines, Tsinghua University Press, Beijing, 2010, (In Chinese).

[36]

Z. Wang, H. Zhang, J. Feng and S. Lun, Present situation and prospect on leak detection and localization techniques for long distance fluid transport pipeline, Control and Instruments in Chemical Industry, 30 (2003), 5-10.

[37]

S. Woon, V. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamics and Systems Theory, 10 (2010), 175-188.

[38]

S. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594. doi: 10.1002/oca.1015.

[39]

K. Yiu, K. Mak and K. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148. doi: 10.3934/jimo.2005.1.133.

[40]

C. Yu, B. Li, R. Loxton and K. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7.

show all references

References:
[1]

N. Ahmed and K. Teo, Optimal Control of Distributed Parameter Systems, North Holland, 1981.

[2]

S. Anita, V. Arnautu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-8098-5.

[3]

V. Arnautu and P. Neittaanmaki, Optimal Control from Theory to Computer Programs, Kluwer Academic, Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3.

[4]

P. Barooah, P. Mehta and J. Hespanha, Mistuning-based control design to improve closed-loop stability margin of vehicular platoons, IEEE Transactions on Automatic Control, 54 (2009), 2100-2113. doi: 10.1109/TAC.2009.2026934.

[5]

S. Blazic, D. Matko and G. Geiger, Simple model of a multi-batch driven pipeline, Mathematics and Computers in Simulation, 64 (2004), 617-630. doi: 10.1016/j.matcom.2003.11.013.

[6]

F. Bullo, J. Cortes and S. Martinez, Distributed Control of Robotic Networks (In Applied Mathematics Series), Princeton University Press, New York, 2009.

[7]

M. Chen and D. Georges, Nonlinear optimal control of an open-channel hydraulic system based on an infinite-dimensional model, in Proceeding of the Conference on Decision and Control, vol. 5, 1999.

[8]

H. Cho and G. Hwang, Optimal design for dynamic spectrum access in cognitive radio networks under rayleigh fading, Journal of Industrial and Management Optimization, 8 (2012), 821-840. doi: 10.3934/jimo.2012.8.821.

[9]

E. Chow, L. Hendrix, M. Herberg, S. Itoh, B. Kong, M. Lall and P. Srevens, Pipeline Politics in Asia: The Intersection of Demand, Energy Markets, and Supply Routes, National Bureau of Asian Research, 2010.

[10]

Y. Ding and S. Wang, Optimal control of open-channel flow using adjoint sensitivity analysis, Journal of Hydraulic Engineering-ASCE, 132 (2006), 1215-1228. doi: 10.1061/(ASCE)0733-9429(2006)132:11(1215).

[11]

Z. Feng, K. Teo and V. Rehbock, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. doi: 10.3934/jimo.2005.1.499.

[12]

Z. Feng, K. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303. doi: 10.1016/j.automatica.2007.09.024.

[13]

G. Ferrari-Trecate, A. Buffa and M. Gati, Analysis of coordination in multi-agent systems through partial difference equations, IEEE Transactions on Automatic Control, 51 (2006), 1058-1063. doi: 10.1109/TAC.2006.876805.

[14]

P. Frihauf and M. Krstic, Leader-enabled deployment onto planar curves: A pde-based approach, IEEE Transactions on Automatic Control, 56 (2011), 1791-1806. doi: 10.1109/TAC.2010.2092210.

[15]

R. Glowinski, J. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, (Encyclopedia of Mathematics and its Applications) Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511721595.

[16]

H. Hao and P. Barooah, On achieving size-independent stability margin of vehicular lattice formations with distributed control, IEEE Transactions on Automatic Control, 57 (2012), 2688-2694. doi: 10.1109/TAC.2012.2191179.

[17]

H. Hao, P. Barooah and P. Mehta, Stability margin scaling laws for distributed formation control as a function of network structure, IEEE Transactions on Automatic Control, 56 (2011), 923-929. doi: 10.1109/TAC.2010.2103416.

[18]

J. Kim, K. Kim, V. Natarajan, S. Kelly and J. Bentsman, PdE-based model reference adaptive control of uncertain heterogeneous multiagent networks, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1152-1167. doi: 10.1016/j.nahs.2008.09.008.

[19]

J. Kim, V. Natarajan, S. Kelly and J. Bentsman, Disturbance rejection in robust PdE-based MRAC laws for uncertain heterogeneous multiagent networks under boundary reference, Nonlinear Analysis: Hybrid Systems, 4 (2010), 484-495. doi: 10.1016/j.nahs.2009.11.005.

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Phaladelphia, 2008. doi: 10.1137/1.9780898718607.

[21]

Z. Lin, Distributed Control and Analysis of Coupled Cell Systems, VDM Verlag, Germany, 2008.

[22]

W. Litvinov, Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions, Journal of Industrial and Management Optimization, 7 (2011), 291-315. doi: 10.3934/jimo.2011.7.291.

[23]

M. Liu, S. Zang and D. Zhou, Fast leak detection and location of gas pipelines based on an adaptive particle filter,, International Journal of Applied Mathematics and Computer Science, 15 (). 

[24]

M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks (In Applied Mathematics Series), Princeton University Press, New York, 2010.

[25]

T. Meurer and M. Krstic, Finite-time multi-agent deployment: A nonlinear pde motion planning approach, Automatica, 47 (2011), 2534-2542. doi: 10.1016/j.automatica.2011.08.045.

[26]

S. Moura and H. Fathy, Optimal boundary control & estimation of diffusion-reaction PDEs, in Proceeding of the Conference on Decision and Control, 2011, 921-928.

[27]

R. Murray, Recent research in cooperative control of multi-vehicle systems,, Journal of Dynamical Systems, (): 571. 

[28]

R. Olfati-Saber and R. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.

[29]

P. Parfomak, Pipeline Safety and Security: Federal Programs, Congress Research Services (CRS) Report for Congress, Washington, DC, 2008.

[30]

M. Rafiee, Q. Wu and A. Bayen, Kalman filter based estimation of flow states in open channels using Lagrangian sensing, Proceedings of the Conference on Decision and Control, (2009), 8266-8271. doi: 10.1109/CDC.2009.5400661.

[31]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks, (Communications and Control Engineering Series) Springer-Verlag, London, 2011.

[32]

A. Sarlette and R. Sepulchre, A PDE viewpoint on basic properties of coordination algorithms with symmetries, in Proceedings of the Conference on Decision and Control, 2009, 5139-5144. doi: 10.1109/CDC.2009.5400570.

[33]

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Edition, SIAM, Philadephia, 2004. doi: 10.1137/1.9780898717938.

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (Graduate Studies in Mathematics), American Mathematical Society, New York, 2010.

[35]

G. Wang and H. Ye, Leakage Detection and Localization of Long Distance Fluid Pipelines, Tsinghua University Press, Beijing, 2010, (In Chinese).

[36]

Z. Wang, H. Zhang, J. Feng and S. Lun, Present situation and prospect on leak detection and localization techniques for long distance fluid transport pipeline, Control and Instruments in Chemical Industry, 30 (2003), 5-10.

[37]

S. Woon, V. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamics and Systems Theory, 10 (2010), 175-188.

[38]

S. Woon, V. Rehbock and R. Loxton, Towards global solutions of optimal discrete-valued control problems, Optimal Control Applications and Methods, 33 (2012), 576-594. doi: 10.1002/oca.1015.

[39]

K. Yiu, K. Mak and K. Teo, Airfoil design via optimal control theory, Journal of Industrial and Management Optimization, 1 (2005), 133-148. doi: 10.3934/jimo.2005.1.133.

[40]

C. Yu, B. Li, R. Loxton and K. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7.

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